Count Ways to Score in a Game

Statement

Suppose there is a game where a player can score either $1$, $2$, or $4$ points in each turn. Given a total score, $n$, find all the possible ways in which you can score these $n$ points.

Note: You may assume that you can use a specific score as many times as you want. Additionally, the order in which we select scores from the list is significant.

Let's say the total points to be earned are $3$. A player can score this total in the following three ways:

• $1, 1$and $1$, in three turns: $1+1+1 = 3$.

• $1$ and then a $2$, in two turns: $1 + 2 = 3$.

• $2$ and then a $1$, in two turns: $2 + 1 = 3$ ...